Unwanted white space in parentheses

I'm trying to write an equation and I used left and right to adjust parentheses. Unfortunately, there is too much white space in the top of parentheses.

Is there any way to reduce the white space?

my equation is:

$$a_n=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\left[\frac{1}{4}\cos(3x)+\frac{3}{4}\cos(x)\right]\cos(nx)\text{d}x=\frac{1}{4}\left[\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(3x)\cos(nx)\text{d}x}_{\text{By lemma: }=\begin{cases}0,& \ n\ne 3 \\ 1,& n=3 \end{cases}}+3\cdot{\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(x)\cos(nx)\text{d}x}_{\text{By lemma: }=\begin{cases}0,& n \ne 1 \\ 1,& n=1 \end{cases}}}\right]$$

and the output is: • Off-topic; Don't use $$...$$. Use $...$ – user31729 Oct 26 '15 at 9:22
• there are three ways: you could raise the expression so it is vertically centred or lower the brackets so they are not vertically centred. But probably better you could just bracket the terms above the underbraces, not the whole term including the brace comments. If you said which you wanted, someone would say how to do it:-) – David Carlisle Oct 26 '15 at 9:24

One possibility is to add \vcenter{\hbox{$\displaystyle$}}:

$$a_n=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\left[\frac{1}{4}\cos(3x)+\frac{3}{4}\cos(x)\right]\cos(nx)\text{d}x=% \frac{1}{4}\left[\vcenter{\hbox{\displaystyle\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(3x)\cos(nx)\text{d}x}_{\text{By lemma: }=% \begin{cases}0,& \ n\ne 3 \\ 1,& n=3 \end{cases}}+3\cdot{\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(x)\cos(nx)\text{d}x}_{\text{By lemma: }=\begin{cases}0,& n \ne 1 \\ 1,& n=1 \end{cases}}}}}\right]$$

Perhaps not a pretty answer, but I've found it works whenever I need it.

• @Galc127 perhaps prettier if you use the code '$$a_n=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\left[\frac{1}{4}\cos(3x)+\frac{3}{4}\cos(x)\right]\cos(nx)\text{d}x=% \frac{1}{4}\underbrace{\left[\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(3x)\cos(nx)\text{d}x \right. }_{\text{By lemma: }=% \begin{cases}0,& \ n\ne 3 \\ 1,& n=3 \end{cases}}+3\cdot{\underbrace{\left.\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(x)\cos(nx)\text{d}x\right]}_{\text{By lemma: }=\begin{cases}0,& n \ne 1 \\ 1,& n=1 \end{cases}}}$$' – Fredrik Johansson Oct 26 '15 at 9:43

May be it looks better like this and perhaps this can be broken in to two lines.

\documentclass{article}
\usepackage{amsmath}
\newcommand*{\myd}{\mathrm{d}}
\begin{document}
$a_n =\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\biggl[\frac{1}{4}\cos(3x)+\frac{3}{4}\cos(x)\biggr] \cos(nx)\myd x =\frac{1}{4}\Biggl[\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(3x)\cos(nx)\myd x}_{\text{By lemma: }=\begin{cases}0,& n\ne 3 \\ 1,& n=3 \end{cases}}+3\cdot{\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(x)\cos(nx)\myd x}_{\text{By lemma: }=\begin{cases}0,& n \ne 1 \\ 1,& n=1 \end{cases}}}\Biggr]$
%
\begin{aligned} a_n &=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\biggl[\frac{1}{4}\cos(3x)+\frac{3}{4}\cos(x)\biggr] \cos(nx)\myd x \\ &=\frac{1}{4}\Biggl[\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(3x)\cos(nx)\myd x}_{\text{By lemma: }=\begin{cases}0,& n\ne 3 \\ 1,& n=3 \end{cases}}+3\cdot{\underbrace{\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\cos(x)\cos(nx)\myd x}_{\text{By lemma: }=\begin{cases}0,& n \ne 1 \\ 1,& n=1 \end{cases}}}\Biggr] \end{aligned}

\end{document} Note the use of \Biggl etc from amsmath instead of dynamic \left and \right. Also the d in dx should by \mathrm not \text.

• dx should probably have a \, in front of it. Looks better – daleif Oct 26 '15 at 12:32